Study of Jordan quasigroups and their construction
Jordan quasigroups are commutative quasigroups satisfying the identity $x^{2}(yx)=(x^{2}y)x$. In this paper we discuss the basic properties of Jordan quasigroups and prove that (i) every commutative idempotent quasigroup is Jordan quasigroup, (ii) if a Jordan quasigroup Q is distributive then Q is i...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Taylor & Francis Group
2018-03-01
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| Series: | Journal of Taibah University for Science |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1080/16583655.2018.1451061 |
| _version_ | 1818913031031095296 |
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| author | Amir Khan Muhammad Shah Hidayat Ullah Khan Gul Zaman |
| author_facet | Amir Khan Muhammad Shah Hidayat Ullah Khan Gul Zaman |
| author_sort | Amir Khan |
| collection | DOAJ |
| description | Jordan quasigroups are commutative quasigroups satisfying the identity $x^{2}(yx)=(x^{2}y)x$. In this paper we discuss the basic properties of Jordan quasigroups and prove that (i) every commutative idempotent quasigroup is Jordan quasigroup, (ii) if a Jordan quasigroup Q is distributive then Q is idempotent, (iii) an idempotent entropic quasigroup satisfies Jordan's identity, (iv) a unipotent quasigroup Q satisfying Jordan's identity satisfies left nuclear square property, (vi) if a quasigroup satisfies LC identity, then alternativity ⇔ Jordan's identity, (vii) for a unipotent Jordan quasigroup Q, $x^{3}y=y^{3}x\ \forall \ x,y\in Q$ and (viii) every Steiner quasigroup is Jordan quasigroup. We also prove some properties of the autotopism of Jordan loops. Moreover, we construct an infinite family of nonassociative Jordan quasigroups whose smallest member is of order 6. |
| first_indexed | 2024-12-19T23:24:01Z |
| format | Article |
| id | doaj.art-f55b151e6a7b4ccb9d37410a026a6706 |
| institution | Directory Open Access Journal |
| issn | 1658-3655 |
| language | English |
| last_indexed | 2024-12-19T23:24:01Z |
| publishDate | 2018-03-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | Journal of Taibah University for Science |
| spelling | doaj.art-f55b151e6a7b4ccb9d37410a026a67062022-12-21T20:01:54ZengTaylor & Francis GroupJournal of Taibah University for Science1658-36552018-03-0112215015410.1080/16583655.2018.14510611451061Study of Jordan quasigroups and their constructionAmir Khan0Muhammad Shah1Hidayat Ullah Khan2Gul Zaman3University of SWATQuaid-e-Azam UniversityUniversity of MalakandUniversity of MalakandJordan quasigroups are commutative quasigroups satisfying the identity $x^{2}(yx)=(x^{2}y)x$. In this paper we discuss the basic properties of Jordan quasigroups and prove that (i) every commutative idempotent quasigroup is Jordan quasigroup, (ii) if a Jordan quasigroup Q is distributive then Q is idempotent, (iii) an idempotent entropic quasigroup satisfies Jordan's identity, (iv) a unipotent quasigroup Q satisfying Jordan's identity satisfies left nuclear square property, (vi) if a quasigroup satisfies LC identity, then alternativity ⇔ Jordan's identity, (vii) for a unipotent Jordan quasigroup Q, $x^{3}y=y^{3}x\ \forall \ x,y\in Q$ and (viii) every Steiner quasigroup is Jordan quasigroup. We also prove some properties of the autotopism of Jordan loops. Moreover, we construct an infinite family of nonassociative Jordan quasigroups whose smallest member is of order 6.http://dx.doi.org/10.1080/16583655.2018.1451061Jordan loopsJordan's identityJordan quasigroupunipotent quasigroup and entropic quasigroup |
| spellingShingle | Amir Khan Muhammad Shah Hidayat Ullah Khan Gul Zaman Study of Jordan quasigroups and their construction Journal of Taibah University for Science Jordan loops Jordan's identity Jordan quasigroup unipotent quasigroup and entropic quasigroup |
| title | Study of Jordan quasigroups and their construction |
| title_full | Study of Jordan quasigroups and their construction |
| title_fullStr | Study of Jordan quasigroups and their construction |
| title_full_unstemmed | Study of Jordan quasigroups and their construction |
| title_short | Study of Jordan quasigroups and their construction |
| title_sort | study of jordan quasigroups and their construction |
| topic | Jordan loops Jordan's identity Jordan quasigroup unipotent quasigroup and entropic quasigroup |
| url | http://dx.doi.org/10.1080/16583655.2018.1451061 |
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